Design of a Simplified Car Suspension System with Matlab - Assignment Example

Design of a (Simplified) Car Suspension System With Matlab Institution Affiliation Student’s Name Date Design of a (Simplified) Car Suspension System The car suspension system is an example of linear time invariant system filter. In a linear produces a sinusoidal output of the same frequency with its sinusoidal input. Each sinusoidal output is normally scaled in a different way and then subjected to a phase change. This output will not contain any additional sinusoidal components that are not in the input. For this reason a linear time invariant thus behave as a filter. It simply changes the amplitude and alters the frequencies’ phase components that are not present in the input. 2 When moving on a smooth surface, the spring becomes slightly compressed. Lets assume this slight compression is represented by  and applying Hooke’s law with the spring stiffness as  then the spring force acting on the mass upwards in this case is given by . and the other force acting downwards is the weight of the chassis i.e. W=. The free body diagram can be represented as below ;   Applying Newton’s second law of motion,  and it should be noted that noting that at this condition, the chassis is at rest i.e.  then  Let the wheel to be displaced upward from the surface of height  which causes the mass (chassis) to be move upwards a distance, h . Because of this movement, the spring will be compressed by a length,  . Hooke’s law can be applied at the spring and this will produce a reaction force that is equal to . And the free body diagram can be represented as in figure below;   Mg Applying the Newton’s second law Mg Rearranging ;  And considering the typical road condition as sinusoidal then,  where H is the maximum displacement of the wheel. The above differential equation then becomes; a) The frequency response Sinusoidal input linear time invariant gives sinusoidal output as shown below Differentiating with respect to t for this equation we have Differentiating further  ** Substituting the expression in the linear constant coefficient equation ** we get; Factoring out the common terms and then rearranging, this becomes; b) The frequency response  is a scalar multiple of the input signal. It thus scales down the amplitude of the input. It can be deduced from the above equation that this is a scaler function of frequency of vibration .Therefore the amplitude depends on the frequency. The answer for this is yes. Since by having a look at the frequency response, Multiplying by  in both numerator and the denominator This is the frequency that will cause a great concern. 3) The customer complains of a bumpy ride is simply because the nature of the surface has changed in such a way that there is increase in the output frequency. From the analyses above, this output is still sinusoidal. Yes with this kind of a system I would recommend changing of the systems parameters in order to achieve a system that will cut down the input vibrations to a comfortable level. 4 a) The new linear constant‐coefficient differential equation is as shown, The introduced term  results from the shear forces from the fluid that is moving between the fixed and the moving component making up the shock absorber. From Newton’s law of viscous fluid, We can get the expression of shear stress in the fluid between the moving plate and the fixed plate as;     b) The frequency response can be calculated s below Input y(t)= | Since  C) In order to utilize the frequencys function, we will need to convert the frequency response into a lapse form This can be done by squaring the denominator and this will give; Then taking components in the denominator, Multiplying by  all through we have, Taking square root of -1 w will have; Substituting  ;  Let H=1 then d) for a unit response for different values damping ratio c) For different value of natural frequency d) 5 a) cut-off frequency is the frequency at which the frequency is equal to , and it is related to the natural frequency as follows. The cut off frequency is usually controlled by the damping constant and spring constant . Spring constant determines the natural frequency of the system and the damping constant determines the damping factor. Cut-off frequency is a corner frequency where an amplitude having higher frequency than the input frequency are reduced. Due to this reason, the ride feels smooth. In order to reduce this cut-off frequency, either the natural frequency or the damping factor should be reduced .By doing so, the step response becomes rapid and this can be seen from step-response graph which shows that, low natural frequency values and lower values makes the system assume stability in a rapid manner. A hard suspension system is one with lower cut-off frequencies such that the time-response of a unit step input gives a rapid respond. Conversely, a soft system has high value that makes the response in a way, sluggish. b) In order to reduce the cut-off frequency, the damping factor should be decreased and this result in a decreased value of the frequency response, thus, the ride feels smooth when cut-off frequency is decreased. This case however, makes the step response sluggish as can be noted from the step response of various damping factors. 6. From this analysis, it can be realized that lower values of damping factor as well as high values of natural frequency should be desired when designing a suspension system of any vehicle. So as to achieve this, a spring of high spring constant should be considered and damper of high damping constant should also be considered. Also care should be adhered to while designing this system so as to avoid such cases where the frequency of the forcing parameter equals to the natural frequency. When such situation occurs, it will cause a resonating system that can be very destructive. Summary This analysis considered the design of car suspension system as a filter to the vibrations caused by the rough surface as the car runs over it. The main aim was to eliminate these vibrations of high amplitude. To do this, two simple systems were put in consideration. The first system considered a case where the rapid amplitude is reduced by replacing them with a vibration of scaled amplitude. This system, however, was noted to retain these induced vibrations, and the customer complained of a bumpy ride. In order to solve this, a damper was then introduced and it would be able to control the speed of these amplitudes by means of ‘braking.' This gave the designer a control parameter i.e., cut-off frequency that helped in selecting the operating point of the system. References Dukkipati, R. V. (2010). Mechanical vibrations. Oxford: Alpha Science International Ltd Haykin, S. S., & Van, V. B. (2002). Signals and systems. New York: Wiley Paraskevopoulos, P. N. (2002). Modern control engineering. New York: Marcel Dekker.. . Read More